Throughout history, scientists, philosophers, mathematicians and PhD students lacking funding for actual research have turned to the thought experiment in hopes of discovering something publishable, thereby retaining tenure and/or attracting the admiration of comely undergraduates.

The best thought experiments throw light into dark corners of the universe and also provide other scientists, philosophers, mathematicians and destitute PhD students a way to kill time while waiting for the bus.

Below is a classic thought experiment, pillaged from my book The Geeks' Guide to World Domination (Be Afraid, Beautiful People). I'll post a new thought experiment each day this week.

Achilles and the Tortoise: Zeno’s Paradox

Here’s a classic, pulled straight from the humanities course you took senior year in high school:

Achilles and a tortoise have a race. Achilles, being much the faster, allows the tortoise a 100-yard head start. They start. Of course, because Achilles allowed the turtle to start ahead of him, it takes time for Achilles to reach the tortoise’s starting point. However, the turtle is no longer there—it’s continued ahead and Achilles must again catch up. But, every time Achilles reaches a point the turtle has passed, the turtle has used the time to travel further ahead. Having always to catch up this distance, Achilles will never catch the tortoise!

Right?

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Comments

Comments

Well being that I am new at this I thought I would try and put my in my 2 cents. First off it would seem that some information is missing, at what speed was the tortoise travelling, and then at what speed was Achilles travelling. If they were travelling at the same speed then no he would never be able to catch him, but if the tortoise was only travelling at 1 mph and Achilles at 10 to 15 then obviously he would overtake the tortoise. Also are we only talking about the starting point as that remains static, as Achilles increases speed he will overtake the tortoise's points at an increasing pace until he reached his maximum speed and eventuallly overtakes the tortoise.

Achilles immediately travels 100mph and the turtle travels a constant 1mph. Both trains leave the station at the same time, traveling north on parallel tracks. There is no wind. Low humidity. Negligible altitude. I posit (or, Aristotle posits) that Achilles will never catch the tortoise. This because Achilles has to first travel half the distance to the turtle, and then half the remaining distance, half of that, etc. Because these halfway points shrink toward infinity, but never become zero, he can't ever overtake the tortoise.

Actually I believe Aristotle disposes of this argument by saying that since the movement of Achilles and the tortoise occur within a finite span of time, it is not possible to touch an infinite number of divisions of time (or distance) in a finite amount of time, so therefore it is a construct only of our minds and not of the world itself. In other words, the divisibility of time and distance to infinity are a construct of the mind alone.

I always like to cheat with quantum physics on this. At a certain point, the distance 'half' would be less than the minimal quantum one can travel. So that that point, Achilles traverses the absolute minimum (quantum) increment, is suddenly ahead of the tortoise, and life goes on. Beyond a quantum length you simply can't divide further. So if the quantum for movement is large (say, 1 Achille's stride, or 1 yard), then any distance less than 1 yard is always immediately traversable.

Where does halving the distance play any part of this? The description of the event says that the turtle simply has a 100 yd head start before Achilles will begin running, and unless the finish line is at 100.1 yards or so - whatever distance the turtle covers in the time it takes Achilles to run 100 yards - Achilles will pass the turtle if he maintains a speed greater than the turtles.

Don't get me wrong - I love Aristotle and little philosophical puzzles - but as posted, this is an exercise in assumptions.

Well actually Aristotle posits that the reasoning of Zeno of Elea is wrong in book 6 of the Physics where he also presents the well know(in that time) paradox, and he solves the paradox by making the simple distinction that between the infinitely divisible and the infinitely extensible, so that even if something is indeed infinitely divisible it is not of an infinite extent. This means that although we can infinitely divide something if it takes 8 secs to cross 100 meters it will take always 8 secs to cross 100 meters independently of how much you subdivide the movement because the subdivision does not entails a rising in the extension(geometrically 100 meters or whatever distance equals to a sum of infinite segments of space of an infinitely small amount).

With the appearance of infinitesimal calculus it also was proved that this infinite series can converge so that this problem can also be resolved and we can calculate when Achiles will get to the turtle, I don't recall exactly the calculus for I have been out of maths for some years now but I will post it as soon as I can get it.

There's another argument along these lines that Zeno proposes which says:

"Consider, then, an arrow in its flight. At any instant its extremity occupies a definite point in its path. Now, while occupying this position it must be at rest there. But how can a point be motionless and yet in motion at the same time?"

Cool point, Nuno. I bet many of us remember intro Calc, in which we approximated the area under a curve my lining up bar graphs beneath it and summing their area. As the bars got thinner, the approximation's accuracy increased. Newton's genius (okay, I know somebody's gonna dispute Newton's claim to Calculus, so let's just put it out there now), was in describing what happens when these bars get infinitely thin, i.e. the exact area under the curve. So too with Zeno's paradox: despite dividing Achille's distance into infintely small chunks, we remain able to calculate how far he needs to travel in order to overtake the tortoise. (So too with the arrow: the times that the arrow is "at rest" are these infinitely short times—the bar graphs under the curve—that when summed together create movement and distance).

I think one of the problems with these paradoxes, is that there is an assumption that these are supposed to represents "strange" happenings in the real world, but we all know how the real world works. The paradox comes in when we introduce this logical system of mathematics that can apparently construct a scenario that goes against such common sense.

This is where we get into concepts about what represents a line and divisibility and whether time falls into that category (the dichotomies involved). This is why I interpret Aristotle's point to be focused on the fact that these are mental constructs rather than physical representations. So while it is possible for the mind to envision such infinties, they cannot (or do not) occur in the real world. It is this discrepancy which has to be bridged if we are to conclude that mathematics has applicability in the real world.

Good point, Gerhard. It's fun, though, to think of situations/constructs that break our understanding of the "real world". Really, this is what Einstein did with Newton's gravity. Newton's G almost worked, but not when stretched to its extremes. This is what Aristotle does with his paradox: stretches our understanding of the world and forces us to revise our constructs. Because mathematics (or science or philosophy or whatever) are stretched by these seeming paradoxes, we work to resolve them, and because we work to resolve them we understand the world a bit better...thanks to the modern magic of ScientificBlogging.com (patent pending).

You manage what you measure. To begin with, the choice of measuring the "remaining distance between achilles and the tortoise" puts an upper limit to what the BEST case scenario would be, ie the distance would at best zero. By design the process cannot even entertain the possibility that achilles would actually OVERTAKE the tortoise, so you know right away there's a serious problem with how the pb is framed.

Also a good example of an even more violent sort of combat than "ultimate fighting"..."infinite fighting"

Here's my belated - and very tongue-in cheek - two cents worth.
(I dispute that there is a missing apostrophe in cents!)
Archimedes solved this paradox of Zeno of Alea by means of a simple experiment. He observed a flaw in the experimental setup. Achilles might be so focused on reaching the finishing line as to trip over the tortoise. To avoid this possibility, Archemedes conducted the experiment as a demonstration of the relationship between the volume of a fluid and its weight. First Achilles, and then the tortoise ran over the same track under the same meteorological conditions. A clepsydra was used to determine the winner. It was found that the number of drops of water for the case of the tortoise, inferred from the weight of the water captured, was far greater than was the case for Archimedes.
Archimedes determined Hercules' velocity to be a value which, converted into the standard S.I. units with which we are so familiar, amounts to over one kilometer per cubic decameter, whereas the velocity of the tortoise was a meagre cubic meter per millimeter.